Finite Group Databasebeta

← #16950327

Subgroups — #16950327

Order 4352.

468 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762608C212yesmaximal
2#14528780C212yesmaximal
3#14526911C212yesmaximal
4C4×C17×C4⋊Q8C211yesmaximal, characteristic
5#11762833—22no—
6C2×#10587507C2214yes—
7#11762608—24no—
8#11762578—22no—
9C2×C17×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8×C17C411yescharacteristic
11C2×C4×Q8×C17C2211yescharacteristic
12C17×#391C411yescharacteristic
13C17×#391C2211yescharacteristic
14C17×#389—22no—
15C17×#384C411yescharacteristic
16C4×C4⋊C4×C17—22no—
17C43×C17C411yescharacteristic
18C22×Q8×C17C2×C411yescharacteristic
19C22×Q8×C17C2311yescharacteristic
20C17×C4⋊Q8—22no—
21C17×C4⋊Q8D412yes—
22C17×C4⋊Q8Q812yes—
23C4×Q8×C17—41no—
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C4×Q8×C17—41no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D412yes—
33C2×C42×C17—21no—
34C2×C42×C17—21no—
35C2×C42×C17—21no—
36C2×C42×C17C2×C411yescharacteristic
37C2×C42×C17C2×C411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17D412yes—
40C2×#63223—24no—
41C17×#95—22no—
42C17×#95—22no—
43#10587507—24no—
44#10587507—24no—
45C4⋊#3622—42no—
46C4⋊#3622—42no—
47C2×Q8×C17—21no—
48C2×Q8×C17—41no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17C4⋊C412yes—
53C2×Q8×C17C2×D411yescharacteristic
54C2×Q8×C17C2×Q811yescharacteristic
55C2×Q8×C17#4112yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D412yes—
65C4⋊C4×C17—24no—
66C4⋊C4×C17—24no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—44no—
71C4⋊C4×C17—42no—
72C42×C17—24no—
73C42×C17—22no—
74C42×C17—22no—
75C42×C17—21no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—41no—
79C42×C17—42no—
80C42×C17—41no—
81C42×C17—41no—
82C42×C17C4⋊C412yes—
83#63223—44no—
84#63223—44no—
85C2×#3622—44no—
86#12802—171nomaximal, Sylow
87C23×C17C2×C22⋊C411yescharacteristic
88Q8×C17—41no—
89Q8×C17—41no—
90Q8×C17—42no—
91Q8×C17—42no—
92Q8×C17—42no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17C2×C4⋊C411yescharacteristic
116C2×C4×C17#11711yescharacteristic
117C2×C4×C17#12112yes—
118C2×C4×C17C22⋊Q812yes—
119#3622—84no—
120C4×C4⋊Q8D1711yescharacteristic
121C2×#356—172no—
122#1801—172no—
123#1394—172no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17#38011yescharacteristic
127C22×C17C2×#9911yescharacteristic
128C22×C17#41011yescharacteristic
129C4×C17—24no—
130C4×C17—22no—
131C4×C17—21no—
132C4×C17—21no—
133C4×C17—22no—
134C4×C17—21no—
135C4×C17—42no—
136C4×C17—42no—
137C4×C17—41no—
138C4×C17—42no—
139C4×C17—81no—
140C4×C17#35612yescomplemented
141C2×C4⋊Q8C2×D1711yescharacteristic
142C2×C4×Q8Dic1711yescharacteristic
143C2×C4×Q8C2×D1711yescharacteristic
144#423—342no—
145C2×#101—174no—
146#391Dic1711yescharacteristic
147#391C2×D1711yescharacteristic
148#389—22no—
149#384Dic1711yescharacteristic
150C4×C4⋊C4—22no—
151C43Dic1711yescharacteristic
152#356—344no—
153#343—342no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17#179911yescharacteristic
157C2×C17C2×#35611yescharacteristic
158C2×C17#213211yescharacteristic
159C22×Q8C2×Dic1711yescharacteristic
160C22×Q8C22×D1711yescharacteristic
161C2×M16—344no—
162C4⋊Q8—22no—
163C4⋊Q8Dic3412yes—
164C4⋊Q8D6812yes—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×Dic1711yescharacteristic
173C2×C4⋊C4C2×Dic1711yescharacteristic
174C2×C4⋊C4C22⋊D1712yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×Dic1711yescharacteristic
179C2×C42C2×Dic1711yescharacteristic
180C2×C42C2×Dic1711yescharacteristic
181C2×C42C22⋊D1712yes—
182C4⋊C8—682no—
183C4⋊C8—682no—
184#101—344no—
185#101—344no—
186#95—22no—
187#95—22no—
188C17#1280211yescharacteristic, complemented, Sylow
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8C4⋊Dic1712yes—
195C2×Q8C2×Dic3411yescharacteristic
196C2×Q8C2×D6811yescharacteristic
197C2×Q8#6325212yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4C22⋊Dic1712yes—
203C22×C4C22⋊Dic1711yescharacteristic
204C22×C4C22⋊Dic1711yescharacteristic
205C22×C4C22×Dic1711yescharacteristic
206C22×C4C2×C22⋊D1712yes—
207M16—684no—
208M16—684no—
209C2×C8—684no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42C4⋊Dic1712yes—
228C23C2×C22⋊Dic1711yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4#1058750511yescharacteristic
255C2×C4#1058750511yescharacteristic
256C2×C4#1058761012yes—
257C2×C4C2×C4⋊Dic1711yescharacteristic
258C2×C4#1058761211yescharacteristic
259C2×C4#1058761612yes—
260C8—1364no—
261C22—21no—
262C22—21no—
263C22#1176266911yescharacteristic
264C22C2×#1058750511yescharacteristic
265C22#1176278311yescharacteristic
266C4—24no—
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4#1176260812yescomplemented
278C2—21no—
279C2—21no—
280C2#1452877211yescharacteristic
281C2C2×#1176260811yescharacteristic
282C2#1453046911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC4×C17×C4⋊Q8C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenterC4×C4⋊Q8D17
hypocenterC17#12802
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
140. C4×C17#356semidirect
188. C17#12802semidirect
277. C4#11762608semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117626082
#145287802
#145269112
C4×C17×C4⋊Q82
#1280217